Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity
Open accessAbstract
We construct quantum circuits that exactly encode the spectra of correlated electron models up to errors from rotation synthesis.By invoking these circuits as oracles within the recently introduced "qubitization" framework, one can use quantum phase estimation to sample states in the Hamiltonian eigenbasis with optimal query complexity Oðλ=ϵÞ, where λ is an absolute sum of Hamiltonian coefficients and ϵ is the target precision.For both the Hubbard model and electronic structure Hamiltonian in a second quantized basis diagonalizing the Coulomb operator, our circuits have T-gate complexity O(N þ logð1=ϵÞ), where N is the number of orbitals in the basis.This scenario enables sampling in the eigenbasis of electronic structure Hamiltonians with T complexity O(N 3 =ϵ þ N 2 logð1=ϵÞ=ϵ).Compared to prior approaches, our algorithms are asymptotically more efficient in gate complexity and require fewer T gates near the classically intractable regime.Compiling to surface code fault-tolerant gates and assuming per-gate error rates of one part in a thousand reveals that one can error correct phase estimation on interesting instances of these problems beyond the current capabilities of classical methods using only about a million superconducting qubits in a matter of hours.
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Using Babbush et al.'s QROM read, each table lookup has a Toffoli count and measurement depth of 2^{c_mul+c_exp} and uses a negligible O(c_mul+c_exp) ancillae.
“Using Babbush et al.'s QROM read (section 3A of [2]), each table lookup has a Toffoli count and measurement depth of 2 c mul + c exp and uses a negligible O ( c mul + c exp ) ancillae.”
From How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits (2021) · cites this paper as [2]
Section III A describes L−1 AND computations for unary iteration and log L ancillae; Section III C applies that technique to QROM lookup. This supports linear lookup cost and logarithmic ancilla use when L = 2^(c_mul+c_exp). The copy gives a T-gate count of 4L−4, however, not the claimed exact Toffoli count or measurement depth. (LLM, conf=0.84)
Experimental — accuracy not yet measuredCrosson et al. demonstrated that this effect is common even for instances that do not exhibit noticeable symmetries.
“Crosson et al. 298 demonstrated that this effect is common even for instances that do not exhibit noticeable symmetries.”
From Quantum Chemistry in the Age of Quantum Computing (2019) · cites this paper as [298]
It is not always necessary to reach the adiabatic regime to achieve a significant overlap with the ground state.
“Note that it is not always necessary to reach the adiabatic regime to achieve a significant overlap with the ground state.”
From Quantum Chemistry in the Age of Quantum Computing (2019) · cites this paper as [298]
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- C. Gidney and R. Babbush, Quantum Read-Only Memory for Implement Efficient Fault-Tolerant Quantum Oracles (in press).
- C. Gidney, R. Babbush, M. Mohseni, and H. Neven, Quantum Read-Only Memory for Implementing Efficient Fault-Tolerant Quantum Oracles (in press).
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Self-citationEvaluating Energy Differences on a Quantum Computer with Robust Phase Estimation
Russo, A., Rudinger, K., Morrison, B., & Baczewski, A. (2021) · Physical Review Letters · doi:10.1103/physrevlett.126.210501 · Cited by 52 · see citing works
Elfving, V. E., Broer, B. W., Webber, M., Gavartin, J. L., Halls, M. D., Lorton, K. P., & Bochevarov, A. D. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2009.12472 · Cited by 93 · see citing works
Power-optimal, stabilized entangling gate between trapped-ion qubits
Blümel, R., Grzesiak, N., Pisenti, N. C., Wright, K., & Nam, Y. (2021) · npj Quantum Information · doi:10.1038/s41534-021-00489-w · Cited by 55 · see citing works
Compilation by stochastic Hamiltonian sparsification
Ouyang, Y., White, D. R., & Campbell, E. T. (2020) · Quantum · doi:10.22331/q-2020-02-27-235 · Cited by 56 · see citing works
Fast Multiqubit Gates through Simultaneous Two-Qubit Gates
Gu, X. Q., Fernández-Pendás, J., Vikstål, P., Abad, T., Warren, C., Bengtsson, A., Tancredi, G., Shumeiko, V. S., Bylander, J., Johansson, G., et al. (2021) · PRX Quantum · doi:10.1103/prxquantum.2.040348 · Cited by 49 · see citing works
Linear-depth quantum circuits for multiqubit controlled gates
da Silva, A. J., & Park, D. K. (2022) · Physical Review A · doi:10.1103/physreva.106.042602 · Cited by 50 · see citing works
Subsystem codes with high thresholds by gauge fixing and reduced qubit overhead
Higgott, O., & Breuckmann, N. P. (2021) · Physical Review X · doi:10.1103/physrevx.11.031039 · Cited by 48 · see citing works
Block-encoding structured matrices for data input in quantum computing
Sünderhauf, C., Campbell, E. T., & Camps, J. (2024) · Quantum · doi:10.22331/q-2024-01-11-1226 · Cited by 50 · see citing works
Hybridized Methods for Quantum Simulation in the Interaction Picture
Rajput, A., Roggero, A., & Wiebe, N. (2022) · Quantum · doi:10.22331/q-2022-08-17-780 · Cited by 46 · see citing works
Self-citationNear- and long-term quantum algorithmic approaches for vibrational spectroscopy
Sawaya, N. P. D., Paesani, F., & Tabor, D. P. (2021) · Physical Review A · doi:10.1103/physreva.104.062419 · Cited by 47 · see citing works
Concrete quantum cryptanalysis of binary elliptic curves
Banegas, G., Bernstein, D. J., van Hoof, I., & Lange, T. (2020) · IACR Transactions on Cryptographic Hardware and Embedded Systems · doi:10.46586/tches.v2021.i1.451-472 · Cited by 48 · see citing works
Quantum unary approach to option pricing
Ramos-Calderer, S., Pérez-Salinas, A., García-Martín, D., Bravo-Prieto, C., Cortada, J., Planagumà, J., & Latorre, J. I. (2021) · Physical Review A · doi:10.1103/physreva.103.032414 · Cited by 47 · see citing works
Analyzing Prospects for Quantum Advantage in Topological Data Analysis
Berry, D. W., Su, Y., Gyurik, C., King, R., Basso, J., Barba, A. D. T., Rajput, A., Wiebe, N., Dunjko, V., & Babbush, R. (2024) · PRX Quantum · doi:10.1103/prxquantum.5.010319 · Cited by 50 · see citing works
Self-citationNam, Y., & Maslov, D. A. (2019) · npj Quantum Information · doi:10.1038/s41534-019-0152-0 · Cited by 41 · see citing works
Warren, C., Fernández-Pendás, J., Ahmed, S., Abad, T., Bengtsson, A., Biznárová, J., Debnath, K., Gu, X. Q., Križan, C., Osman, A., et al. (2023) · npj Quantum Information · doi:10.1038/s41534-023-00711-x · Cited by 41 · see citing works
Compilation by stochastic Hamiltonian sparsification
Ouyang, Y., White, D. R., & Campbell, E. T. (2020) · White Rose Research Online (University of Leeds, The University of Sheffield, University of York) · Cited by 41 · see citing works
Preparing Bethe Ansatz Eigenstates on a Quantum Computer
Dyke, J. S. V., Barron, G. S., Mayhall, N. J., Barnes, E., & Economou, S. E. (2021) · PRX Quantum · doi:10.1103/prxquantum.2.040329 · Cited by 44 · see citing works
Rapid Initial-State Preparation for the Quantum Simulation of Strongly Correlated Molecules
Berry, D. W., Tong, Y., Khattar, T., White, A. F., Kim, T. I., Low, G. H., Boixo, S., Ding, Z., Lin, L., Lee, S., et al. (2025) · PRX Quantum · doi:10.1103/prxquantum.6.020327 · Cited by 41 · see citing works
Self-citationNo-Go Theorems for Quantum Resource Purification: New Approach and Channel Theory
Fang, K., & Liu, Z.-W. (2022) · PRX Quantum · doi:10.1103/prxquantum.3.010337 · Cited by 41 · see citing works
Estimating Quantum Speedups for Lattice Sieves
Albrecht, M. R., Gheorghiu, V., Postlethwaite, E. W., & Schanck, J. M. (2020) · International Conference on the Theory and Application of Cryptology and Information Security · doi:10.1007/978-3-030-64834-3_20 · Cited by 58 · see citing works
Quantum circuit optimization with AlphaTensor
Ruiz, F. J. R., Laakkonen, T., Bausch, J., Balog, M., Barekatain, M., Heras, F. J. H., Novikov, A., Fitzpatrick, N., Romera‐Paredes, B., van de Wetering, J., et al. (2025) · Nature Machine Intelligence · doi:10.1038/s42256-025-01001-1 · Cited by 44 · see citing works
- Authors
- Babbush, R., Gidney, C., Berry, D. W., Wiebe, N., McClean, J., Paler, A., Fowler, A., & Neven, H.
- Affiliations
- Google (United States) (US)
- Macquarie University (AU)
- Microsoft (United States) (US)
- ORCID
- Venue
- Physical Review X
- Publisher
- American Physical Society (APS)
- Published
- 2018-10-23
- Volume
- 8
- Issue
- 4
- ISSN
- 2160-3308 (ISSN-L)
- OpenAlex
- W2801305581
- Cited by
- 383
- References
- 118
- Citations by year
- 2026: 45 · 2025: 82 · 2024: 52 · 2023: 49 · 2022: 43 · 2021: 41 · 2020: 35 · 2019: 32 · 2018: 4
- FWCI
- 15.90
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Licence
- Locations
- Physical Review X (publishedVersion, cc-by)
- arXiv (Cornell University) (submittedVersion)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
- arXiv (Cornell University)
- Funders
- Australian Research Council (DP160102426)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Keywords
- quantum circuits, qubitization, quantum phase estimation, Hubbard model, superconducting qubits, query complexity, correlated electrons
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
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